The Adjoint Spectral Green’s Function Method Applied to Direct and Inverse Neutral Particle Source–Detector Problems

  • Jesús P. Curbelo
  • Odair P. da Silva
  • Ricardo C. Barros


In direct source–detector problems the use of the adjoint technique allows to obtain the detector response due to multiple sources by a single solution to the adjoint problem in each energy group. On the other hand, in inverse source–detector problems it is possible to calculate the intensity of the source in each energy group given its location and the detector response. This work is based on the application of the adjoint spectral Green’s function method (SGF) for solving direct and inverse source–detector transport problems in the energy multigroup discrete ordinates formulation with arbitrary L′th-order of scattering anisotropy. The offered SGF method along with the one-region block inversion iterative scheme generates numerical solutions that are completely free from spatial truncation errors; therefore, a spatial reconstruction scheme is developed to analytically determine the detector response in direct problems and source intensities in inverse problems.



This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001. The authors also acknowledge the partial financial support of Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro—Brasil (FAPERJ) and Conselho Nacional de Desenvolvimento Científico e Tecnológico—Brasil (CNPq).


  1. [EnEtAl96]
    Engl, H. W., Hanke, M., and Neubauer, A.: Regularization of Inverse Problems, Kluwer Academic Publishers, Dordrecht, The Netherlands (1996).CrossRefGoogle Scholar
  2. [Al94]
    Alifanov, O. M.: Inverse Heat Transfer Problems, Springer–Verlag, Berlin Heidelberg (1994).Google Scholar
  3. [MoSi13]
    Moura Neto, F. D., and Silva Neto, A. J.: An Introduction to Inverse Problems with Applications, Springer–Verlag, Berlin Heidelberg (2013).CrossRefGoogle Scholar
  4. [HyAz11]
    Hykes, J. M., and Azmy, Y. Y.: Radiation source reconstruction with known geometry and materials using the adjoint. In International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering (M&C) 2011, Latin American Section (LAS) / American Nuclear Society (ANS), Rio de Janeiro, Brazil (2011).Google Scholar
  5. [MiEtAl12]
    Militão, D. S., Alves, H. and Barros, R. C.: A numerical method for monoenergetic slab–geometry fixed-source adjoint transport problems in the discrete ordinates formulation with no spatial truncation error. International Journal of Nuclear Energy Science and Technology, 7, 151–165 (2012).CrossRefGoogle Scholar
  6. [CuEtAl17]
    Curbelo, J. P., da Silva, O. P., García, C. R., and Barros, R. C.: Shifting Strategy in the Spectral Analysis for the Spectral Green’s Function Nodal Method for Slab–Geometry Adjoint Transport Problems in the Discrete Ordinates Formulation. In Integral Methods in Science and Engineering, Volume 2: Practical Applications, C. Constanda et al. (eds.), Birkhäuser Basel (2017), Ch. 20, pp. 201–210.Google Scholar
  7. [CuEtAl18]
    Curbelo, J. P., da Silva, O. P., and Barros, R. C.: An adjoint technique applied to slab–geometry source–detector problems using the generalized spectral Green’s function nodal method. Journal of Computational and Theoretical Transport, (2018). (doi:10.1080/23324309.2018.1539403)MathSciNetCrossRefGoogle Scholar
  8. [Mc92]
    McCormick, N. J.: Inverse Radiative Transfer Problems: A Review. Nuclear Science and Engineering, 112, 185–198 (1992).CrossRefGoogle Scholar
  9. [DuMa79]
    Duderstadt, J. J. and Martin, W. R.: Transport Theory. Wiley–Interscience, New York, USA, (1979).Google Scholar
  10. [PrLa10]
    Prinja, A. K. and Larsen, E. W.: General Principles of Neutron Transport. Cacuci, D. G. (Ed), Handbook of Nuclear Engineering, Ch. 5. Springer Science+Business Media, New York, USA (2010).Google Scholar
  11. [LeMi93]
    Lewis, E. E. and Miller, W. F.: Computational methods of neutron transport. American Nuclear Society, Illinois, USA, (1993).Google Scholar
  12. [GaSi83]
    Garcia, R. D. M. and Siewert, C. E.: Multislab multigroup transport theory with L′th order anisotropic scattering. Journal of Computational Physics, 50, 181–192 (1983).CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Jesús P. Curbelo
    • 1
  • Odair P. da Silva
    • 1
  • Ricardo C. Barros
    • 1
  1. 1.Instituto PolitécnicoUniversidade do Estado do Rio de JaneiroNova FriburgoBrazil

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